How to merge a loose edge into coplanar faces?
AFAIK, there is no way to use an edge (without even a plane) to divide a face.

What you need to do is delete the edge (x->edge) and then join the two verts, by selecting them and hitting 'j'.

If you've made a bunch of edges not realizing what you were doing in some other model, you can use mesh->cleanup->delete loose to delete all these edges, but I don't believe there's any way to transform them into cuts.